QM MAP: Glacier Sensitivity
Shawn Schneidereit
18.4.2021
note that the .Rmd can be found on my personal repo for the corse
Constraining the effects of topography and climate on climate change sensitivity of glaciers in Tibet
The Tibetan Plateau region is home to the greatest amount of non-polar glaciers, which are also highly sensitive to climate change and has recently been experiencing the greatest rates of mass losses. These glaciers are primary freshwater sources to major rivers across Asia, with increased rates of melting increasing the risks of floods and other geo-hazards to some of the world’s most densely populated regions. Yet the geophysical and atmospheric conditions driving changes in Tibetan plateau glacier mass balance remain relatively understudied, with uncertainties in which variables can best predict glacier mass change. In this analysis I will use four different modeling approaches to assess which topographical and climatological drives are most important in explaining changes in glacier equilibrium line altitude (ELA) and whether these drives can be used to explain glacier ELA changes since the little ice age. The first approach uses a Mixed effects model with random effects, where all variables where chosen based on a priori knowledge gained from the readings and an exploratory correlation plot. Following this I constructed an identically structured hierarchical Bayesian model (using the MCMCglm package), to compare the identified importance of variables under a frequentest vs baysian framework. As the exploratory correlation plot showed that many variables displayed high levels of collinearity/multicollinearity, I decided to create a model using principal components from a PCA, to preserve all available variables, while not violating any assumptions of variable collinearity and co-dependence. Finally, I decided to use an automatic variable selection method (MUMIn), where I included the maximum number of variables possible, within the computational limits of my PC. While I believe that the Baysian modeling approaches are best suited to identifying which variables influence ELA and dELA, due to their combination of using pre defined variables chosen based on mechanistic processes and their relatively low BIC values when cross compared. In conclusion I synthesize the results all four models to judge which varibales are most important for predicitng ELA and if ELA shifts since the little ice age can be predicted using topographical and climatological variables.
In this analysis I attempted to justify all modeling structure and variable selection decisions made. As I quite unfamiliar with climatic modeling and glacier dynamics, some of these decisions may be sub-optimal. Yet, these decisions were made to the best of my abilities, without going over the top on background research and overall, my models should have a coherent structure and sufficient work-flow documentation. In none of the preliminary readings, did I encountered the temperature or precipitation in a specific month singled out to be of particular importance for glacier ELA. Therefore, I decided to exclude all monthly climatic variables and only focus on the aggregated factors. Along with the provided variables, I also included a new variable called temperature variability, as my preliminary readings highlighted its potential importance (Brun, et. al., 2017) and I was able to compute it with the provided dataset. While I include p-values throughout my analysis, in this report I interpret variable significance as having a relatively large effect, with a standard error small enough that when ± to the effect size, do not make it cross 0 (the implication being that the effect size is large enough to have a measurable influence, even when considering the uncertainty around the estimate).
1.Data import and wrangeling
glacier <- read_csv("~/Documents/humbolt/semester_01/quantitative_methods/assignments/glacier_sensitivity.csv") %>%
na.exclude ##
## ── Column specification ────────────────────────────────────────────────────────
## cols(
## .default = col_double(),
## orientation = col_character()
## )
## ℹ Use `spec()` for the full column specifications.glacier_scaled <- glacier %>%
# adding a new variable for temperature variability; a factor mentioned to be important in multiple papers
mutate(T_variability = ((T_MAX_mean_monsoon + T_MAX_mean_not_monsoon)-(T_MIN_mean_monsoon + T_MIN_mean_not_monsoon) /2),
# scaling all varibales used in modelling
length = scale(length, center = T, scale = T),
area = scale(area, center = T, scale = T),
P_snow = scale(P_snow, center = T, scale = T),
P_year = scale(P_year, center = T, scale = T),
P_monsoon = scale(P_monsoon, center = T, scale = T),
P_not_monsoon = scale(P_not_monsoon, center = T, scale = T),
T_MIN_mean_monsoon = scale(T_MIN_mean_monsoon, center = T, scale = T),
T_MIN_mean_not_monsoon = scale(T_MIN_mean_not_monsoon, center = T, scale = T),
T_MAX_mean_monsoon = scale(T_MAX_mean_monsoon, center = T, scale = T),
T_MAX_mean_not_monsoon = scale(T_MAX_mean_not_monsoon, center = T, scale = T),
T_variability = scale(T_variability, center = T, scale = T),
T_mean_mea.yr = scale(T_mean_mea.yr, center = T, scale = T),
T_mean_monsoon = scale(T_mean_monsoon, center = T, scale = T),
T_mean_not_monsoon = scale(T_mean_not_monsoon, center = T, scale = T),
Slope_min = scale(Slope_min, center = T, scale = T),
Slope_max = scale(Slope_max, center = T, scale = T),
Slope_mean = scale(Slope_mean, center = T, scale = T),
Elev_min = scale(Elev_min, center = T, scale = T),
Elev_max = scale(Elev_max, center = T, scale = T),
Elev_mean = scale(Elev_mean, center = T, scale = T))
# Here is the for a exploratory correlation plot I used to visually assess my variables to determine what to potentially include in my models, and to identify where strong variable co-dependency existed. I commented out the code as the plot is quite busy and to not detract from the flow of the rest of my analysis.
#(corr_plot <- ggpairs(data = glacier, columns = c(2:9,22:25,38:39,52:53,66:74)))
#corrplot.mixed(cor1[,2:9,22:25,38:39,52:53,66:74], lower.col = "black" , number.cex = .7)
# Here a number of strong correlations between variables can be seen. Notable is this Summit and Max elevation seem to be the same variable and that particular temperature and precipitation variables correlation with each other. Q1. Which are the most important topographical and climatological drivers of glacier equilibrium line altitude (ELA)?
Mixed effects model with random effects
In this first section, I chose to create a Mixed effects model with random effects (lme4 package), where all variables where chosen based on a priori knowledge gained from the readings and the exploratory correlation plot. The methodological advantage of this approach is that through including the random effects, correlations between data coming from specific morphology types and geographical orientations can be accounted for, by treating them as a grouping factor. While this prevents us from explicitly assessing the impacts of morphology and geographic orientation, it still allows us to see how they influence the observed patterns in other variables. In the final section of this analysis (the automated variable selection) morphology and geographic orientation are left as fixed effects, so that I have them once as a fixed and once as a random effect, to allow for comparison between the two scenarios. As the Distribution of ELA had a slight negative skew, I also tried running a GLM with a inverse gaussian distribution (link = “1/mu^2”), but encountered the error that the “PIRLS step-halvings failed to reduce deviance in pwrssUpdate”, to which I did not find a solution.
hist(glacier_scaled$ELA) # there is a negative skew to the data, but overall a gaussian distibution looks appropriate
m_lme4 <- lmer(ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean + (1|morph_type) + (1|orientation),
data = glacier_scaled)
summary(m_lme4)## Linear mixed model fit by REML ['lmerMod']
## Formula:
## ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon +
## T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean +
## (1 | morph_type) + (1 | orientation)
## Data: glacier_scaled
##
## REML criterion at convergence: 9555.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -6.1407 -0.4238 0.0367 0.4765 6.2845
##
## Random effects:
## Groups Name Variance Std.Dev.
## orientation (Intercept) 129.2 11.37
## morph_type (Intercept) 468.9 21.65
## Residual 4819.7 69.42
## Number of obs: 849, groups: orientation, 8; morph_type, 6
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 5098.664 12.038 423.553
## debris_cov 2.774 11.248 0.247
## length -23.132 6.984 -3.312
## area -21.740 6.526 -3.332
## P_snow 20.095 4.940 4.068
## P_monsoon 8.067 4.476 1.802
## P_not_monsoon -33.825 7.117 -4.753
## T_mean_mea.yr -101.070 10.953 -9.227
## T_variability 59.366 7.939 7.477
## Slope_mean -6.581 3.339 -1.971
## Elev_max 8.239 5.691 1.448
## Elev_mean 141.943 7.940 17.876
##
## Correlation of Fixed Effects:
## (Intr) dbrs_c length area P_snow P_mnsn P_nt_m T_mn_. T_vrbl
## debris_cov -0.096
## length -0.059 -0.044
## area -0.031 -0.060 -0.722
## P_snow 0.022 -0.004 -0.039 0.069
## P_monsoon -0.006 0.039 -0.163 0.077 0.016
## P_not_monsn -0.026 0.023 0.126 -0.127 -0.713 -0.546
## T_mean_m.yr 0.024 -0.081 0.021 -0.085 0.026 -0.047 -0.138
## T_variablty 0.008 -0.013 -0.058 0.124 0.256 0.115 -0.116 -0.689
## Slope_mean 0.059 0.005 0.095 0.177 0.066 -0.061 0.025 -0.384 0.177
## Elev_max -0.012 -0.131 -0.319 -0.191 -0.028 0.010 0.090 -0.092 0.165
## Elev_mean 0.031 -0.016 0.160 0.041 -0.109 -0.094 0.132 0.612 -0.105
## Slp_mn Elv_mx
## debris_cov
## length
## area
## P_snow
## P_monsoon
## P_not_monsn
## T_mean_m.yr
## T_variablty
## Slope_mean
## Elev_max -0.477
## Elev_mean -0.067 -0.371The most important fixed effects where mean elevation and mean temperature. A more detailed description and visualization can be found in the dot and whiskar plot at the end of this section. Some variables such a snow and precipitation in non monsoon phases, as well as mean annual temperature and temperature variability snowed some correlation, but as I assess these to have different mechanistic reasons to be included, I decided to not alter my model structure.
# R squared
r.squaredGLMM(m_lme4)## Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.## R2m R2c
## [1,] 0.8996553 0.9107324Here we mainly need to focus on the The R²(C), which is 90.78 %. The C represents the Conditional R_GLMM² and can be interpreted as a variance explained by the entire model, including both fixed and random effects
# checking out variable co-linearity
plot(m_lme4)
The residual vs fitted plot shows a mostly flat and even distribution of points, with greater distances seen at the lower spectrum of points, likely due to the negative skew described above
qqnorm(resid(m_lme4))
qqline(resid(m_lme4))
When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Ideal would be to use a distribution that better fits the data (such as a inverse gussian distribution), which unfortunately was not possible. Following an ad hoc transformation of the dependent variable ELA, this observed trend only showed a very marginal improvement, so I decided to keep ELA in its un-transformed state.
# random effects
plot_model(m_lme4, type = "re", show.values = TRUE, vline.color = "black")## [[1]]
##
## [[2]]
In the summary it can be seen that overall both random effects orientation and morphology type account for relatively large amounts of variation, with Morphology being the stronger predictor (variance 468,9; std dev. 21.65), vs orientation (variance 129.2, std dev. 11.37). Yet in the visualization of the random effect we can see that that for neither orientation or morphology, is are any individual groups with a large outlying effect, as all effect sizes ± standard errors either are very close or cross 0.
ggcoefstats(x = m_lme4, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F) 
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can clearly see that mean elevation and mean annual temperature are the two most influential variables at predicting ELA. These are then followed by precipitation in non-monsoon phases, as well as the temperature variability variable that I aggregated from mean temperature data.
Mixed effects model with random effects dELA
hist(glacier_scaled$dELA)
There is a stronger skew to the data. While again an inverse Gaussian distribution seems appropriate, this was also not possible. Instead I compared a the model with a normal Gaussian and a Poisson distribution and found that the Poisson had a better distribution of residuals.
m_lme4 <-lmer(ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean + (1 | morph_type) + (1 | orientation),
data = glacier_scaled)
m_lme4_d <-glmer(dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean + (1 | morph_type) + (1 | orientation),
family = poisson,
data = glacier_scaled)
summary(m_lme4_d)## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: poisson ( log )
## Formula:
## dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon +
## T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean +
## (1 | morph_type) + (1 | orientation)
## Data: glacier_scaled
##
## AIC BIC logLik deviance df.resid
## 26667.4 26733.8 -13319.7 26639.4 835
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -9.4921 -3.6701 -0.7578 2.7556 30.8456
##
## Random effects:
## Groups Name Variance Std.Dev.
## orientation (Intercept) 0.01299 0.1140
## morph_type (Intercept) 0.67998 0.8246
## Number of obs: 849, groups: orientation, 8; morph_type, 6
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.297871 0.342767 12.539 < 2e-16 ***
## debris_cov -0.287961 0.016273 -17.696 < 2e-16 ***
## length 0.022230 0.009986 2.226 0.026009 *
## area -0.004885 0.009229 -0.529 0.596575
## P_snow 0.034998 0.007133 4.907 9.27e-07 ***
## P_monsoon 0.047429 0.006502 7.294 3.00e-13 ***
## P_not_monsoon -0.104107 0.010147 -10.260 < 2e-16 ***
## T_mean_mea.yr 0.230208 0.014964 15.384 < 2e-16 ***
## T_variability -0.022877 0.010742 -2.130 0.033192 *
## Slope_mean 0.091553 0.004734 19.339 < 2e-16 ***
## Elev_max 0.059419 0.007670 7.747 9.41e-15 ***
## Elev_mean 0.038329 0.010282 3.728 0.000193 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) dbrs_c length area P_snow P_mnsn P_nt_m T_mn_. T_vrbl
## debris_cov -0.002
## length -0.007 -0.040
## area -0.002 -0.071 -0.706
## P_snow 0.004 -0.015 -0.039 0.075
## P_monsoon 0.002 0.052 -0.184 0.067 0.010
## P_not_monsn -0.005 0.028 0.151 -0.135 -0.718 -0.552
## T_mean_m.yr 0.001 -0.093 -0.020 -0.080 0.011 -0.036 -0.105
## T_variablty 0.000 -0.017 -0.047 0.125 0.254 0.168 -0.175 -0.736
## Slope_mean 0.003 0.005 0.119 0.167 0.078 -0.101 0.023 -0.419 0.215
## Elev_max -0.001 -0.105 -0.322 -0.194 -0.034 0.047 0.063 -0.049 0.129
## Elev_mean 0.000 -0.052 0.134 0.040 -0.149 -0.039 0.145 0.605 -0.181
## Slp_mn Elv_mx
## debris_cov
## length
## area
## P_snow
## P_monsoon
## P_not_monsn
## T_mean_m.yr
## T_variablty
## Slope_mean
## Elev_max -0.482
## Elev_mean -0.097 -0.359In the summary it can be seen that neither random effects orientation and morphology type have significant influence on ELA, with Morphology as both have relatively small effect sizes with standard errors that make the estimate cross 0. The most important fixed effects where mean temperature and debris cover. A more detailed description and visualization can be found in the dot and whisker plot at the end of this section. Some variables such a snow and precipitation in non monsoon phases, as well as mean annual temperature and temperature variability snowed some correlation, but as I assess these to have different mechanistic reasons to be included, I decided to not alter my model structure.
# R squared (conditional)
r.squaredGLMM(m_lme4_d)## Warning: The null model is correct only if all variables used by the original
## model remain unchanged.## R2m R2c
## delta 0.04899643 0.9864697
## lognormal 0.04899972 0.9865360
## trigamma 0.04899310 0.9864028The R²(C) is 98.65 %, where the C represents the Conditional R_GLMM² and can be interpreted as a variance explained by the entire model, including both fixed and random effects
# checking out variable co-linearity
plot(m_lme4_d)
The residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermediate values
qqnorm(resid(m_lme4_d))
When checking the QQ-plot it is seen that there is slight tail at the upper bound, but overall this is a significant improvement over the Gaussian distribution.
# random effects
plot_model(m_lme4_d, type = "re", show.values = TRUE, vline.color = "black")## [[1]]
##
## [[2]]
In the summary it can be seen that both random effects orientation and morphology type relatively account for a small amount of overall variation, with Morphology being the stronger predictor (variance 0.68; std dev. 0.82), vs orientation (variance 0.012, std dev. 0.11). Note that here the random effects are not centered around 0 but 1, as I used a Possion distribution for this model. In the visualization of the random effect we can see that that for orientation there are no groups with a large outlying effect, as all effect sizes ± standard errors either are very close or cross 0. On the other hand, for morphology type we can see that class 6 has a uncharacteristically strong negative influence on dELA, indicating that it may be a driver of glacier loss.
# Variable effect size visualization
ggcoefstats(x = m_lme4_d, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F) 
In the dot-whiskar visual representation of the effect sizes and standard errors of our the variables we can clearly see that as before (normal ELA) mean annual temperature is an highly influential variable at predicting ELA. This is followed by followed by the previously identified variable of precipitation in non-monsoon phases, as well as the new debris cover variable and mean elevation.
Bayesian Models
MCMCglmm is an altnerative modeling package to Rstan (brms), that I selected to use as I am more family with its practical implementation and as I have been unable to install stan successfully in the past. Both packages use a Bayesian statistical framework in combination with Markov chain Monte Carlo simulations. The default prior values assumed in a MCMCglmm are posterior distribution with very large variance values for the fixed effects and a flat (weakly informative) prior. The variances for the random effects are assumed to have inverse-Wishart priors with a very low value for nu (weakly informative). I decided to not delve deeper into assigning my own priors, due to me having i) little prior knowledge the dynamics / expected values and distributions of variables and ii) relatively shallow knowledge of the mathematical underpinnings and practical implementation methods for assigning more informative priors.
m_bays <- MCMCglmm(ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean, random = ~ morph_type + orientation,
data = glacier_scaled) ## Warning: Unknown or uninitialised column: `family`.## Warning: Setting row names on a tibble is deprecated.##
## MCMC iteration = 0
##
## MCMC iteration = 1000
##
## MCMC iteration = 2000
##
## MCMC iteration = 3000
##
## MCMC iteration = 4000
##
## MCMC iteration = 5000
##
## MCMC iteration = 6000
##
## MCMC iteration = 7000
##
## MCMC iteration = 8000
##
## MCMC iteration = 9000
##
## MCMC iteration = 10000
##
## MCMC iteration = 11000
##
## MCMC iteration = 12000
##
## MCMC iteration = 13000# The model runs and converges after the default 13000 iterations
summary(m_bays)##
## Iterations = 3001:12991
## Thinning interval = 10
## Sample size = 1000
##
## DIC: 9644.451
##
## G-structure: ~morph_type
##
## post.mean l-95% CI u-95% CI eff.samp
## morph_type 661.8 1.937e-10 2409 584.9
##
## ~orientation
##
## post.mean l-95% CI u-95% CI eff.samp
## orientation 85.99 1.063e-16 344.2 310.4
##
## R-structure: ~units
##
## post.mean l-95% CI u-95% CI eff.samp
## units 4896 4432 5370 502.8
##
## Location effects: ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean
##
## post.mean l-95% CI u-95% CI eff.samp pMCMC
## (Intercept) 5099.632 5073.123 5123.610 804.6 <0.001 ***
## debris_cov 5.187 -18.061 25.510 1000.0 0.646
## length -24.554 -37.676 -10.694 894.5 <0.001 ***
## area -22.418 -35.541 -11.543 1000.0 <0.001 ***
## P_snow 19.550 9.746 28.685 1000.0 <0.001 ***
## P_monsoon 9.726 -0.658 18.180 280.0 0.050 .
## P_not_monsoon -33.734 -47.420 -19.942 898.6 <0.001 ***
## T_mean_mea.yr -95.052 -117.068 -69.611 152.8 <0.001 ***
## T_variability 56.454 41.028 72.149 100.9 <0.001 ***
## Slope_mean -7.296 -13.988 -0.882 1000.0 0.028 *
## Elev_max 9.233 -1.399 20.548 1000.0 0.112
## Elev_mean 145.283 127.854 161.175 176.5 <0.001 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As before in the hierarchical GLM, it is seen that both morphology and orientation account for some variance in ELA. From the postierier mean (visualization in following code chunk), we can see that that like in the hierarchical GLM, mean elevation and mean annual temperature are the strongest predictors of ELA (as its postierier mean in centered farthest from zero). The aggregated variable temperature variability was the also a good predictor of ELA.
# R squared
#r.squaredGLMM(m_bays)Unfortunately I was not able to figure out access using the r2 with a predefined function, or an official way to compute it. Therefore For me Bayesian models I do not include r2 values.
# postirior distribution of random effects
par(mfrow = c(1,2))
hist(mcmc(m_bays$VCV)[,"morph_type"])
hist(mcmc(m_bays$VCV)[,"orientation"])
The variance can not be equal to zero, but as the mean value is up against zero, this can be interpreted as the effect not being of great significance (which corresponds with the results of the hierarchical linear model). As the spread of the histograms are relatively narrow, it can be assumed that the distribution is relatively accurate.
# Assesing model convergence through plotting trace and density est. for the intercept
plot(m_bays$Sol)
The trace can be interpreted as the a pseudo time-series of what the model was doing in each of its iterations. As all traces have the “fuzzy caterpillar” appearance, it can be assumed there is adequate mixing with overall convergence being met. The density plot shows the postierier distribution of each the model predicted for each variable in each iteration. If the distribution crosses zero, the variable can be assumed to be non-significant and the spread of the distribution conveys the overall accuracy of the predicted variable, as well as the probability of effect size being “true”.
# Variable effect size visualization
ggcoefstats(x = m_bays, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", robust = TRUE, exclude.intercept = T)## Warning: Can't extract 'deviance' residuals. Returning response residuals.
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can clearly see that as before (Hierarchical GLM), the mean elevation and mean annual temperature are highly influential variables at predicting ELA. This is followed by followed by the also previously identified variables of temperature variability and precipitation in non-monsoon phases. The correspondence between the hierarchical GLM and Bayesian model in terms of the magnitude of how variables influence ELA makes intuitive sense, as largely the models largely share the same hierarchical structure and only slightly deviate in how effect sizes and standard errors are calculated. The lack of major differences between the two models is also accentuated by the fact the only weak (non-informative) priors were selected for the Bayesian model.
Bayesian dELA
m_bays_d <- MCMCglmm(dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean, random = ~ morph_type + orientation,
family = "poisson",
data = glacier_scaled) ##
## MCMC iteration = 0
##
## Acceptance ratio for liability set 1 = 0.000220
##
## MCMC iteration = 1000
##
## Acceptance ratio for liability set 1 = 0.441896
##
## MCMC iteration = 2000
##
## Acceptance ratio for liability set 1 = 0.455330
##
## MCMC iteration = 3000
##
## Acceptance ratio for liability set 1 = 0.466432
##
## MCMC iteration = 4000
##
## Acceptance ratio for liability set 1 = 0.509494
##
## MCMC iteration = 5000
##
## Acceptance ratio for liability set 1 = 0.509485
##
## MCMC iteration = 6000
##
## Acceptance ratio for liability set 1 = 0.509521
##
## MCMC iteration = 7000
##
## Acceptance ratio for liability set 1 = 0.509187
##
## MCMC iteration = 8000
##
## Acceptance ratio for liability set 1 = 0.510649
##
## MCMC iteration = 9000
##
## Acceptance ratio for liability set 1 = 0.509330
##
## MCMC iteration = 10000
##
## Acceptance ratio for liability set 1 = 0.510311
##
## MCMC iteration = 11000
##
## Acceptance ratio for liability set 1 = 0.510081
##
## MCMC iteration = 12000
##
## Acceptance ratio for liability set 1 = 0.510303
##
## MCMC iteration = 13000
##
## Acceptance ratio for liability set 1 = 0.510100# The model runs and converges after the default 13000 iterations
summary(m_bays_d)##
## Iterations = 3001:12991
## Thinning interval = 10
## Sample size = 1000
##
## DIC: 7017.478
##
## G-structure: ~morph_type
##
## post.mean l-95% CI u-95% CI eff.samp
## morph_type 0.002907 3.561e-17 0.001499 1000
##
## ~orientation
##
## post.mean l-95% CI u-95% CI eff.samp
## orientation 0.007013 1.699e-16 0.02354 309.8
##
## R-structure: ~units
##
## post.mean l-95% CI u-95% CI eff.samp
## units 0.292 0.2641 0.3234 847.5
##
## Location effects: dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean
##
## post.mean l-95% CI u-95% CI eff.samp pMCMC
## (Intercept) 4.471700 4.396902 4.551156 1000.0 <0.001 ***
## debris_cov -0.194139 -0.360286 -0.033519 1000.0 0.024 *
## length 0.018940 -0.091080 0.124414 1000.0 0.672
## area -0.057611 -0.171369 0.032298 1000.0 0.254
## P_snow 0.029123 -0.041720 0.114626 1000.0 0.462
## P_monsoon 0.069296 0.002700 0.149383 677.9 0.058 .
## P_not_monsoon -0.125731 -0.240085 -0.013529 836.9 0.034 *
## T_mean_mea.yr 0.382773 0.207533 0.572074 428.3 <0.001 ***
## T_variability -0.078725 -0.213707 0.044303 699.7 0.202
## Slope_mean 0.077949 0.021030 0.129824 1000.0 0.002 **
## Elev_max 0.070856 -0.007976 0.162023 901.7 0.104
## Elev_mean 0.127486 0.003874 0.245267 586.0 0.032 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As before in the hierarchical GLM, it is seen that both morphology and orientation account for some variance in ELA. From the postierier mean (visualization in following code chunk), we can see that that like in the hierarchical GLM, mean annual temperature is the strongest predictor of ELA (as its postierier mean in centered farthest from zero). Additionally mean elevation, debris cover, and precipitation during non monsoon periods are also seen to be influential variables. A more detailed description can be found at the end of the section in the dot whiskar plot.
# R squared
#r.squaredGLMM(m_bays_d)Unfortunately I was not able to figure out access using the r2 with a predefined function, or an official way to compute it. Therefore For me Bayesian models I do not include r2 values.
# posterior distribution of random effects
par(mfrow = c(1,2))
hist(mcmc(m_bays_d$VCV)[,"morph_type"])
hist(mcmc(m_bays_d$VCV)[,"orientation"])
The variance can not be equal to zero, but as the mean value is up against zero, this can be interpreted as the effect not being of great significance. As the spread of the histograms are relatively narrow, it can be assumed that the distribution is relatively accurate.
# Assessing model convergence through plotting trace and density est. for the intercept
plot(m_bays_d$Sol)
The trace can be interpreted as the a pseudo time-series of what the model was doing in each of its iterations. As all traces have the fuzzy caterpillar appearance, it can be assumed there is adequate mixing with overall convergence being met. The density plot shows the postierer distribution of each the model predicted for each variable in each iteration. If the distribution crosses zero, the variable can be assumed to be non-significant and the spread of the distribution conveys the overall accuracy of the predicted variable, as well as the probability of effect size being “true”. .
# Variable effect size visualization
ggcoefstats(x = m_bays_d, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", robust = TRUE, exclude.intercept = T)## Warning: Can't extract 'deviance' residuals. Returning response residuals.
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can clearly see that as before (Hierarchical GLM), the mean annual temperature and debris cover are highly influential variables at predicting dELA. This is followed by followed by the also previously identified variable of precipitation in non-monsoon phases. The correspondence between the hierarchical GLM and Bayesian model in terms of the magnitude of how variables influence dELA makes intuitive sense, as largely the models largely share the same hierarchical structure and only slightly deviate in how effect sizes and standard errors are calculated. The lack of major differences between the two models is also accentuated by the fact the only weak (non-informative) priors were selected for the Bayesian model.
PCA
In an exploratory correlation plot I found that many variables displayed high levels of collinearity/multicollinearity. Therefore, I decided to create a model using the principal components from a PCA, which has the advantage of being able to preserve all available variables, while not violating any assumptions of variable collinearity and co-dependence. Using the rotation function to assess how strongly each variable loaded on to the computed PC, I created a characterization of the which variables each PC was analogous to. I selection critireon for the number of PC included in the final model was a visual assessment of the cumulative sum of explained variation, relying on the “elbow-method”.
# pca # need to redo analysis/description of pca plots
pca <- glacier_scaled %>%
dplyr::select(c(4:9,22:25,38:39,52:53,66:75), -orientation) # need to remove as it is a character
str(pca)## tibble[,23] [849 × 23] (S3: tbl_df/tbl/data.frame)
## $ summit : num [1:849] 5710 5750 5715 5760 5675 ...
## $ debris_cov : num [1:849] 0 0 0 0 0 0 0 0 0 0 ...
## $ morph_type : num [1:849] 2 2 2 3 2 2 2 2 2 2 ...
## $ length : num [1:849, 1] -0.386 -0.284 -0.548 -0.579 0.295 ...
## ..- attr(*, "scaled:center")= num 6216
## ..- attr(*, "scaled:scale")= num 5539
## $ area : num [1:849, 1] -0.3439 -0.2619 -0.4021 -0.4901 0.0947 ...
## ..- attr(*, "scaled:center")= num 1393269
## ..- attr(*, "scaled:scale")= num 2117729
## $ P_snow : num [1:849, 1] -0.5453 -0.2486 -0.0552 0.9086 0.1799 ...
## ..- attr(*, "scaled:center")= num 120
## ..- attr(*, "scaled:scale")= num 50.7
## $ P_year : num [1:849, 1] -1.211 -0.811 -0.735 -0.782 -0.027 ...
## ..- attr(*, "scaled:center")= num 889
## ..- attr(*, "scaled:scale")= num 177
## $ P_monsoon : num [1:849, 1] -1.121 -0.593 -0.599 -0.564 0.481 ...
## ..- attr(*, "scaled:center")= num 638
## ..- attr(*, "scaled:scale")= num 94.4
## $ P_not_monsoon : num [1:849, 1] -1.162 -0.936 -0.788 -0.912 -0.538 ...
## ..- attr(*, "scaled:center")= num 251
## ..- attr(*, "scaled:scale")= num 93.3
## $ T_MIN_mean_monsoon : num [1:849, 1] -1.18 -1.59 -1.41 -1.72 -1.43 ...
## ..- attr(*, "scaled:center")= num 1.13
## ..- attr(*, "scaled:scale")= num 1.08
## $ T_MIN_mean_not_monsoon: num [1:849, 1] -1.24 -1.61 -1.43 -1.7 -1.47 ...
## ..- attr(*, "scaled:center")= num -11.9
## ..- attr(*, "scaled:scale")= num 1.3
## $ T_MAX_mean_monsoon : num [1:849, 1] -1.08 -1.59 -1.38 -1.75 -1.36 ...
## ..- attr(*, "scaled:center")= num 7.81
## ..- attr(*, "scaled:scale")= num 0.892
## $ T_MAX_mean_not_monsoon: num [1:849, 1] -1.16 -1.62 -1.41 -1.76 -1.46 ...
## ..- attr(*, "scaled:center")= num -2.75
## ..- attr(*, "scaled:scale")= num 1.04
## $ T_mean_mea.yr : num [1:849, 1] -1.18 -1.61 -1.42 -1.74 -1.45 ...
## ..- attr(*, "scaled:center")= num -2.4
## ..- attr(*, "scaled:scale")= num 1.08
## $ T_mean_monsoon : num [1:849, 1] -1.13 -1.6 -1.41 -1.74 -1.4 ...
## ..- attr(*, "scaled:center")= num 4.47
## ..- attr(*, "scaled:scale")= num 0.983
## $ T_mean_not_monsoon : num [1:849, 1] -1.21 -1.61 -1.43 -1.74 -1.47 ...
## ..- attr(*, "scaled:center")= num -7.31
## ..- attr(*, "scaled:scale")= num 1.16
## $ Slope_min : num [1:849, 1] 0.69002 -0.16035 0.41288 0.00442 -0.79263 ...
## ..- attr(*, "scaled:center")= num 8.45
## ..- attr(*, "scaled:scale")= num 6.45
## $ Slope_max : num [1:849, 1] -0.138 -0.748 -0.6 -0.855 -1.09 ...
## ..- attr(*, "scaled:center")= num 46.2
## ..- attr(*, "scaled:scale")= num 10.4
## $ Slope_mean : num [1:849, 1] -0.359 -0.925 -0.342 -0.431 -1.285 ...
## ..- attr(*, "scaled:center")= num 27.5
## ..- attr(*, "scaled:scale")= num 6.44
## $ Elev_min : num [1:849, 1] 0.837 0.901 1.071 1.336 0.896 ...
## ..- attr(*, "scaled:center")= num 4800
## ..- attr(*, "scaled:scale")= num 407
## $ Elev_max : num [1:849, 1] 0.641 0.86 0.699 0.991 0.548 ...
## ..- attr(*, "scaled:center")= num 5505
## ..- attr(*, "scaled:scale")= num 292
## $ Elev_mean : num [1:849, 1] 0.994 1.377 1.464 1.71 1.191 ...
## ..- attr(*, "scaled:center")= num 5155
## ..- attr(*, "scaled:scale")= num 221
## $ T_variability : num [1:849, 1] -0.929 -1.527 -1.286 -1.73 -1.291 ...
## ..- attr(*, "scaled:center")= num 10.4
## ..- attr(*, "scaled:scale")= num 0.782
## - attr(*, "na.action")= 'exclude' Named int [1:4] 731 739 793 829
## ..- attr(*, "names")= chr [1:4] "731" "739" "793" "829"pca <- prcomp(pca, scale. = T)
summary(pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 3.239 2.1197 1.7267 1.26260 1.03151 0.82279 0.68121
## Proportion of Variance 0.456 0.1954 0.1296 0.06931 0.04626 0.02943 0.02018
## Cumulative Proportion 0.456 0.6514 0.7810 0.85032 0.89658 0.92601 0.94619
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 0.57066 0.52564 0.41415 0.37528 0.32194 0.28148 0.25031
## Proportion of Variance 0.01416 0.01201 0.00746 0.00612 0.00451 0.00344 0.00272
## Cumulative Proportion 0.96035 0.97236 0.97982 0.98594 0.99045 0.99389 0.99662
## PC15 PC16 PC17 PC18 PC19 PC20
## Standard deviation 0.20272 0.15590 0.11041 0.01307 0.006805 0.004639
## Proportion of Variance 0.00179 0.00106 0.00053 0.00001 0.000000 0.000000
## Cumulative Proportion 0.99840 0.99946 0.99999 1.00000 1.000000 1.000000
## PC21 PC22 PC23
## Standard deviation 9.62e-08 9.493e-16 5.442e-16
## Proportion of Variance 0.00e+00 0.000e+00 0.000e+00
## Cumulative Proportion 1.00e+00 1.000e+00 1.000e+00pca$rotation[,1:6]## PC1 PC2 PC3 PC4
## summit -0.1156595114 -0.396530022 0.006722996 -0.1575511536
## debris_cov 0.0592119096 -0.269462680 -0.052365271 0.0841236946
## morph_type 0.0400162434 -0.091990076 -0.029584626 0.0998727770
## length 0.0483474510 -0.419110439 0.066102380 0.2009610901
## area 0.0498102447 -0.403993717 0.069942759 0.2063360677
## P_snow 0.0092334579 0.044215327 0.512080560 -0.1067217199
## P_year 0.1663450945 0.026036585 0.473270085 -0.0983638200
## P_monsoon 0.1149693609 -0.015678340 0.482582772 -0.1269818451
## P_not_monsoon 0.1989915206 0.065216422 0.408839901 -0.0579761041
## T_MIN_mean_monsoon 0.3060541207 0.012778292 -0.053013928 0.0008543518
## T_MIN_mean_not_monsoon 0.3032010943 -0.016791648 -0.044913670 -0.0236876350
## T_MAX_mean_monsoon 0.2981733398 0.062127235 -0.096162241 0.0460564355
## T_MAX_mean_not_monsoon 0.3038968050 0.008468283 -0.074287604 0.0015814723
## T_mean_mea.yr 0.3054852777 0.009768718 -0.064004869 0.0001986579
## T_mean_monsoon 0.3041721614 0.035272945 -0.072790921 0.0211532886
## T_mean_not_monsoon 0.3047792037 -0.005670976 -0.058388518 -0.0124484181
## Slope_min -0.0008673191 0.172349229 -0.144086417 -0.5497905945
## Slope_max 0.1074204433 -0.340938679 -0.052861818 -0.2748845202
## Slope_mean 0.0969690782 -0.129444291 -0.151594973 -0.6427029063
## Elev_min -0.2245285603 0.284072988 0.023014192 -0.0242427371
## Elev_max -0.1272311906 -0.398418302 0.024459017 -0.1469850576
## Elev_mean -0.2903997157 -0.014890301 0.052015325 -0.1132050921
## T_variability 0.2804581157 0.087127203 -0.134391044 0.0736330210
## PC5 PC6
## summit 0.0824178265 -0.013314116
## debris_cov -0.1631885443 -0.921362046
## morph_type -0.8875920506 0.231071412
## length -0.0376147737 0.144502015
## area -0.1262642924 0.122834120
## P_snow -0.1045303257 -0.040817253
## P_year -0.0006254728 -0.028488009
## P_monsoon 0.0538899676 -0.034703229
## P_not_monsoon -0.0557100478 -0.018888474
## T_MIN_mean_monsoon 0.0092062920 -0.004116058
## T_MIN_mean_not_monsoon 0.0024765380 0.003884554
## T_MAX_mean_monsoon 0.0178121599 -0.010498486
## T_MAX_mean_not_monsoon 0.0277241682 -0.001477082
## T_mean_mea.yr 0.0136610122 -0.001637198
## T_mean_monsoon 0.0131736061 -0.006805675
## T_mean_not_monsoon 0.0138881375 0.001490011
## Slope_min -0.3032269288 -0.114844999
## Slope_max 0.1929150277 0.181647499
## Slope_mean -0.0472596556 0.047772626
## Elev_min -0.0550247898 -0.073168174
## Elev_max 0.0741599741 0.021785456
## Elev_mean -0.0209235332 -0.053078593
## T_variability 0.0487193561 -0.014295989Based on this rotation, I will make the following assumption for which variables are analogs to each PC PC1: All temperature variables (both monsoon & non-monsoon), temperature variability, and elevation mean PC2: Geophysical conditions such as min/max elevation, debris cover, catchment length and area, and max slope PC3: All precipitation variables PC4: Slope, and to a far lesser degree catchment area, length and elevation PC5: Morphology type
var_exp <- data.frame(pc = c(1:23),
var_exp = pca$sdev^2 / sum(pca$sdev^2))
# add the variances cumulatively
var_exp$var_exp_cumsum <- cumsum(var_exp$var_exp)
var_exp## pc var_exp var_exp_cumsum
## 1 1 4.560124e-01 0.4560124
## 2 2 1.953597e-01 0.6513721
## 3 3 1.296334e-01 0.7810055
## 4 4 6.931112e-02 0.8503167
## 5 5 4.626144e-02 0.8965781
## 6 6 2.943396e-02 0.9260121
## 7 7 2.017606e-02 0.9461881
## 8 8 1.415870e-02 0.9603468
## 9 9 1.201310e-02 0.9723599
## 10 10 7.457546e-03 0.9798175
## 11 11 6.123134e-03 0.9859406
## 12 12 4.506432e-03 0.9904470
## 13 13 3.444813e-03 0.9938918
## 14 14 2.724236e-03 0.9966161
## 15 15 1.786818e-03 0.9984029
## 16 16 1.056764e-03 0.9994597
## 17 17 5.299700e-04 0.9999896
## 18 18 7.426372e-06 0.9999971
## 19 19 2.013656e-06 0.9999991
## 20 20 9.358421e-07 1.0000000
## 21 21 4.023442e-16 1.0000000
## 22 22 3.917793e-32 1.0000000
## 23 23 1.287748e-32 1.0000000ggplot(var_exp) +
geom_bar(aes(x = pc, y = var_exp), stat = "identity") +
geom_line(aes(x = pc, y = var_exp_cumsum)) +
theme_classic()
Following the Elbow method, it seems like 3-5 PCs seems like an appropriate cutoff point. Despite being more difficult to interpret the final results, I decided to include 5 PCs, due to PCs 4 and 5 increasing the overall variance explained from ~80% to 90%. This also helps to ensure that the large number of variables in the model are captured in the PCs, especially as morphology type almost exclusively loads on PC5.
# scores
pca_scores <- data.frame(pca$x)
head(pca_scores)## PC1 PC2 PC3 PC4 PC5 PC6
## 1 -4.085950 0.12736786 -1.15488972 -0.4276191 0.8461927 -0.19220332
## 2 -5.283278 -0.02000121 0.10140922 0.2929743 0.9265436 -0.25750181
## 3 -4.742488 0.34258840 -0.03086408 -0.5009767 0.7503837 -0.36280031
## 4 -5.739056 0.13114428 0.67963939 -0.2685366 -0.6152369 -0.08876674
## 5 -4.418274 -0.03063813 1.46720119 1.0095502 0.9809866 -0.22197371
## 6 -3.354310 0.37954311 0.86186526 0.4488272 1.1837522 -0.16445739
## PC7 PC8 PC9 PC10 PC11 PC12
## 1 -0.5924383 0.3157648 -0.06914327 0.040799740 -0.33150857 0.003904768
## 2 -0.4260974 -0.0194212 0.07569821 -0.284495967 -0.08137097 -0.125315391
## 3 -0.4281015 0.1150437 0.10357952 0.006878345 0.01024527 -0.130232249
## 4 0.3062982 0.4004047 0.59410478 -0.129178425 0.05062106 0.089638527
## 5 -0.4874418 -0.4850412 -0.13692691 -0.028121838 0.24060972 -0.082023890
## 6 0.2822511 -0.2910021 -0.38920635 -0.053532077 0.18451985 -0.127911367
## PC13 PC14 PC15 PC16 PC17 PC18
## 1 -0.06771404 -0.007749453 0.03161404 -0.08157295 -0.02674952 -0.02366540
## 2 -0.12582001 0.104398650 0.04780135 -0.07585605 -0.01790055 -0.01942436
## 3 -0.16104792 0.015764018 0.06123426 -0.03189586 -0.02538805 -0.01376238
## 4 0.16573447 -0.050168883 0.09239933 -0.16075389 -0.01365135 -0.02749751
## 5 0.19389144 0.139088085 -0.01502534 -0.08903185 0.07761670 -0.01844080
## 6 0.31287542 -0.240958925 -0.19863270 0.03489515 0.11186253 -0.01939790
## PC19 PC20 PC21 PC22 PC23
## 1 0.008769504 0.0026651323 -2.861286e-08 -5.551115e-17 1.387779e-16
## 2 0.001274638 0.0082995976 7.317158e-08 -2.220446e-16 2.220446e-16
## 3 -0.002945277 0.0004759622 -4.582566e-08 2.220446e-16 3.608225e-16
## 4 0.001010093 -0.0047553419 2.704832e-07 -1.110223e-16 3.053113e-16
## 5 0.003278927 -0.0033848377 1.013831e-07 -6.661338e-16 -1.249001e-16
## 6 0.003915545 -0.0039411928 -2.274929e-07 -1.110223e-16 -8.326673e-17pca_scores_df <- cbind(glacier_scaled, pca_scores)
#ggpairs(pca_scores_df[,c(4:6,8:9,22:25,38:39,52:53,66:75,76:80)]) # first 5 Pcs
m_pca <- glm(ELA ~ PC1 + PC2 + PC3 + PC4 + PC5, data = pca_scores_df)
summary(m_pca)##
## Call:
## glm(formula = ELA ~ PC1 + PC2 + PC3 + PC4 + PC5, data = pca_scores_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -365.98 -41.14 -3.62 41.31 562.01
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5104.9305 2.8600 1784.941 < 2e-16 ***
## PC1 -65.1737 0.8836 -73.757 < 2e-16 ***
## PC2 17.7810 1.3500 13.171 < 2e-16 ***
## PC3 4.7192 1.6573 2.848 0.00451 **
## PC4 -24.0376 2.2665 -10.606 < 2e-16 ***
## PC5 -4.2597 2.7743 -1.535 0.12506
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 6944.475)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 5854193 on 843 degrees of freedom
## AIC: 9927.3
##
## Number of Fisher Scoring iterations: 2Here we see that PCs 1,2, and 4 have the strongest effect size and large predictive power for ELA. Based on the characterization of which variables are represented by each PC, this means that temperature, geophysical conditions, and slope (and to a smaller extend catchment length, area, and elevation) are the most important variables. Out of the three PCs, PC1 (temperature variables), are by far the most important with a effect size of -65.17 (Std. error 0.88). PCs 2 (geophysical conditions) and 4 (slope) have effect sizes of 17.78 (Std. error 1.35) and -24.04 (Std. error 2.27) respectively.
# R squared
r.squaredGLMM(m_pca)## R2m R2c
## [1,] 0.8712126 0.8712126# The R² is 87.12 %plot(m_pca)
The residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermediate values. The QQ-plot has skewed tails at both the lower and upper bounds and is substantial for larger values
# Variable effect size visualization
ggcoefstats(x = m_pca, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F)
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can clearly see that PCs 1,2, and 4 have the strongest effect size and large predictive power for ELA. Based on the characterization of which variables are represented by each PC, this means that temperature, geophysical conditions, and slope (and to a smaller extend catchment length, area, and elevation) are the most important variables.
PCA dELA model
m_pca_d <- glm(dELA ~ PC1 + PC2 + PC3 + PC4 + PC5, family = poisson, data = pca_scores_df)
summary(m_pca_d)##
## Call:
## glm(formula = dELA ~ PC1 + PC2 + PC3 + PC4 + PC5, family = poisson,
## data = pca_scores_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -12.3519 -4.1720 -0.8174 2.7615 26.3817
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.5928575 0.0034796 1319.926 <2e-16 ***
## PC1 0.0344310 0.0010108 34.063 <2e-16 ***
## PC2 -0.0290177 0.0015856 -18.301 <2e-16 ***
## PC3 0.0005099 0.0018754 0.272 0.786
## PC4 -0.0952305 0.0025894 -36.777 <2e-16 ***
## PC5 0.0391401 0.0032913 11.892 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 26056 on 848 degrees of freedom
## Residual deviance: 23131 on 843 degrees of freedom
## AIC: 28476
##
## Number of Fisher Scoring iterations: 5Here we see that PCs 4 and 5 have the strongest effect size and large predictive power for dELA. Based on the characterization of which variables are represented by each PC, this means that slope (and to a smaller extend catchment length, area, and elevation), as well as morphological conditions are the most important variables for predicting dELA. Out of the two PCs, PC4 (slope+), is slightly more important with a effect size of -0.095 (Std. error 0.003). PCs 5 (morphology) has an effect size of 0.039 (Std. error 0.0032).
# R squared
r.squaredGLMM(m_pca_d)## Warning: The null model is correct only if all variables used by the original
## model remain unchanged.## R2m R2c
## delta 0.7644043 0.7644043
## lognormal 0.7652960 0.7652960
## trigamma 0.7635058 0.7635058# The R² is 76.44 %plot(m_pca_d)
The residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermediate values
# Variable effect size visualization
ggcoefstats(x = m_pca_d, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F)
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can see that PCs 4 and 5 have the strongest effect size and large predictive power for dELA. Based on the characterization of which variables are represented by each PC, this means that slope (and to a smaller extend catchment length, area, and elevation) and morphology are the most important variables.
Automated variable selection
For the automated variable variable I focused on variables that I previously found to be strong predictors of ELA, aswell as the de-aggregated forms of temperature and precipitation metrics. While potential interaction between temperature and precipitation variables are likely to exist, these were not included in the saturated model, as just including a reduced list of variables with no interactions, already brought my computer to the edges of its computational limits. Overall, variables where selected based on a priori assumptions on what I thought would be will be more relevant to include, while also factoring in the outputs of my previous models and exploratory analysis.
m_saturated <-
glm( ELA ~ debris_cov + morph_type + orientation + length + P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon + T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean, # + T_MIN_mean_monsoon + T_MIN_mean_not_monsoon + T_MAX_mean_monsoon T_MAX_mean_not_monsoon ; taken out due to computational constraints. Additionally, to get the model to run, area, max/mean slope, t_variability and monsoon precipitation also needed to be removed. I chose these variables, as in previous models and in exploratory analysis they tended have low impacts on ELA.
data = glacier_scaled,
family = gaussian)
summary(m_saturated)##
## Call:
## glm(formula = ELA ~ debris_cov + morph_type + orientation + length +
## P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean,
## family = gaussian, data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -319.00 -22.28 -0.55 21.49 273.08
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.087e+03 8.798e+00 578.204 < 2e-16 ***
## debris_cov 1.875e+01 7.928e+00 2.365 0.018255 *
## morph_type -2.507e-01 2.733e+00 -0.092 0.926953
## orientationN 1.495e+01 5.812e+00 2.572 0.010297 *
## orientationNE 1.537e+01 6.553e+00 2.345 0.019255 *
## orientationNW 2.370e+01 7.226e+00 3.280 0.001081 **
## orientationS 2.711e+01 6.794e+00 3.991 7.17e-05 ***
## orientationSE 6.481e+00 8.088e+00 0.801 0.423155
## orientationSW 2.608e+01 7.460e+00 3.496 0.000498 ***
## orientationW 2.775e+01 6.691e+00 4.147 3.72e-05 ***
## length -8.992e+00 3.391e+00 -2.652 0.008158 **
## P_snow 6.372e+00 3.453e+00 1.845 0.065359 .
## P_year 1.664e+01 6.409e+00 2.597 0.009567 **
## P_not_monsoon -3.239e+01 7.727e+00 -4.192 3.06e-05 ***
## T_mean_mea.yr -4.231e+05 1.411e+07 -0.030 0.976093
## T_mean_monsoon 1.597e+05 5.329e+06 0.030 0.976091
## T_mean_not_monsoon 2.646e+05 8.830e+06 0.030 0.976097
## Slope_min 5.444e+00 1.985e+00 2.742 0.006237 **
## Elev_min 1.803e+02 5.926e+00 30.421 < 2e-16 ***
## Elev_max 8.835e+01 4.591e+00 19.244 < 2e-16 ***
## Elev_mean 6.351e+00 7.194e+00 0.883 0.377567
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2401.556)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 1988489 on 828 degrees of freedom
## AIC: 9040.6
##
## Number of Fisher Scoring iterations: 2options(na.action = "na.fail") # Required for dredge to run
# Run all possible model permutations and tank according to AIC values
m_dredge <- dredge(m_saturated, beta = F, evaluate = T, rank = AICc)## Fixed term is "(Intercept)"options(na.action = "na.omit") # set back to default
head(m_dredge)## Global model call: glm(formula = ELA ~ debris_cov + morph_type + orientation + length +
## P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean,
## family = gaussian, data = glacier_scaled)
## ---
## Model selection table
## (Int) dbr_cov Elv_max Elv_men Elv_min lng orn P_not_mns P_snw P_yer
## 10204 5086 19.26 90.23 183.9 -9.266 + -30.19 5.733 15.09
## 9948 5086 19.37 91.32 184.9 -9.292 + -24.51 15.51
## 8156 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 14300 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 12252 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 10208 5086 19.03 88.16 4.689 181.4 -9.070 + -29.66 5.595 14.72
## Slp_min T_men_mea.yr T_men_mns T_men_not_mns df logLik AICc delta
## 10204 5.427 -28.51 18 -4498.965 9034.8 0.00
## 9948 5.395 -30.98 17 -4500.478 9035.7 0.94
## 8156 5.498 -60.91 33.41 19 -4498.702 9036.3 1.57
## 14300 5.498 10.41 -38.10 19 -4498.702 9036.3 1.57
## 12252 5.498 27.58 -55.36 19 -4498.702 9036.3 1.57
## 10208 5.348 -26.91 19 -4498.733 9036.4 1.63
## weight
## 10204 0.291
## 9948 0.182
## 8156 0.133
## 14300 0.133
## 12252 0.133
## 10208 0.129
## Models ranked by AICc(x)nrow(m_dredge) # 16384 models in total## [1] 16384top_model <- get.models(m_dredge, subset = 1)[[1]]
top_model##
## Call: glm(formula = ELA ~ debris_cov + Elev_max + Elev_min + length +
## orientation + P_not_monsoon + P_snow + P_year + Slope_min +
## T_mean_not_monsoon + 1, family = gaussian, data = glacier_scaled)
##
## Coefficients:
## (Intercept) debris_cov Elev_max Elev_min
## 5086.266 19.258 90.232 183.873
## length orientationN orientationNE orientationNW
## -9.266 14.967 15.421 23.393
## orientationS orientationSE orientationSW orientationW
## 27.646 7.058 26.234 27.895
## P_not_monsoon P_snow P_year Slope_min
## -30.188 5.733 15.093 5.427
## T_mean_not_monsoon
## -28.514
##
## Degrees of Freedom: 848 Total (i.e. Null); 832 Residual
## Null Deviance: 45690000
## Residual Deviance: 1992000 AIC: 9034# Summarize top model
summary(top_model)##
## Call:
## glm(formula = ELA ~ debris_cov + Elev_max + Elev_min + length +
## orientation + P_not_monsoon + P_snow + P_year + Slope_min +
## T_mean_not_monsoon + 1, family = gaussian, data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -319.02 -22.06 -0.35 21.54 274.51
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5086.266 4.483 1134.445 < 2e-16 ***
## debris_cov 19.258 7.854 2.452 0.014408 *
## Elev_max 90.232 3.345 26.971 < 2e-16 ***
## Elev_min 183.873 4.494 40.917 < 2e-16 ***
## length -9.266 3.227 -2.871 0.004192 **
## orientationN 14.967 5.797 2.582 0.009994 **
## orientationNE 15.421 6.537 2.359 0.018550 *
## orientationNW 23.393 7.206 3.246 0.001216 **
## orientationS 27.646 6.726 4.110 4.34e-05 ***
## orientationSE 7.058 8.052 0.877 0.381004
## orientationSW 26.234 7.412 3.539 0.000424 ***
## orientationW 27.895 6.616 4.216 2.76e-05 ***
## P_not_monsoon -30.188 6.948 -4.345 1.57e-05 ***
## P_snow 5.733 3.327 1.723 0.085203 .
## P_year 15.093 5.881 2.566 0.010454 *
## Slope_min 5.427 1.950 2.783 0.005505 **
## T_mean_not_monsoon -28.514 4.442 -6.419 2.31e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2393.755)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 1991604 on 832 degrees of freedom
## AIC: 9033.9
##
## Number of Fisher Scoring iterations: 2The most important fixed effects where elevation variables and and the various orientations. A more detailed description and visualization can be found in the dot and whisker plot at the end of this section.
# psudo R^2
r.squaredGLMM(top_model) # 95.74%## R2m R2c
## [1,] 0.9556106 0.9556106plot(top_model)
The residual vs fitted plot shows a mostly flat and even distribution of points, with greater diviances seen at the lower spectrum of points, likely due to the negative skew described above
When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Following an ad hoc transformation of the dependent variable ELA, this observed trend only showed a very marginal improvement, so I decided to keep ELA in its un-transformed state.
# Variable effect size visualization
ggcoefstats(x = top_model, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F)
The automated model section based on AIC ranking of 16384 models, indicated that min and max elevation have are the two most important variables to predict ELA (with effect sizes 183.87 and 90.23 respectively & with small std.errors of 4.49 and 3.34), with large effect sizes relative to other variables. The other next most important variables where the various orientation directions, as well as precipitation and temperature in non monsoon periods. A notable orison is mean yearly temperature, which in the glmer model had the highest effect size. This could be attributed to high co-linearity between mean temperature and temperature in non monsoon periods, but with temperature in non monsoon periods have a large effect size and therefor being prioritized in the selection.
Automated variable selection dELA
m_saturated_d <-
glm( dELA ~ debris_cov + morph_type + orientation + length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon + T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min + Elev_max + Elev_mean, # + T_MIN_mean_monsoon + T_MIN_mean_not_monsoon + T_MAX_mean_monsoon T_MAX_mean_not_monsoon ; taken out due to computational constraints. Additionally, to get the model to run, area, max slope, t_variability snow, and monsoon precipitation also needed to be removed. I chose these variables, as in previous models and in exploratory analysis they tended have low impacts on ELA.
data = glacier_scaled,
family = poisson)
summary(m_saturated_d)##
## Call:
## glm(formula = dELA ~ debris_cov + morph_type + orientation +
## length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min +
## Elev_max + Elev_mean, family = poisson, data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -13.0002 -3.8463 -0.6707 2.7638 23.1736
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.606e+00 1.829e-02 251.793 < 2e-16 ***
## debris_cov -2.407e-01 1.596e-02 -15.084 < 2e-16 ***
## morph_type -2.529e-02 5.659e-03 -4.469 7.84e-06 ***
## orientationN -3.527e-02 1.248e-02 -2.825 0.00473 **
## orientationNE -1.955e-02 1.404e-02 -1.392 0.16378
## orientationNW -1.892e-02 1.560e-02 -1.213 0.22519
## orientationS 2.311e-01 1.363e-02 16.960 < 2e-16 ***
## orientationSE 5.327e-02 1.687e-02 3.158 0.00159 **
## orientationSW 2.150e-01 1.482e-02 14.504 < 2e-16 ***
## orientationW 1.781e-01 1.367e-02 13.029 < 2e-16 ***
## length 5.331e-02 6.913e-03 7.712 1.24e-14 ***
## P_year 3.890e-02 1.308e-02 2.975 0.00293 **
## P_not_monsoon -5.807e-02 1.363e-02 -4.259 2.05e-05 ***
## T_mean_mea.yr 7.064e+04 2.970e+04 2.379 0.01737 *
## T_mean_monsoon -2.667e+04 1.121e+04 -2.379 0.01737 *
## T_mean_not_monsoon -4.419e+04 1.858e+04 -2.379 0.01737 *
## Slope_min -3.609e-02 4.563e-03 -7.909 2.59e-15 ***
## Slope_mean 1.237e-01 5.366e-03 23.052 < 2e-16 ***
## Elev_min 2.705e-01 1.196e-02 22.623 < 2e-16 ***
## Elev_max 1.673e-01 9.120e-03 18.340 < 2e-16 ***
## Elev_mean -1.754e-01 1.372e-02 -12.788 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 26056 on 848 degrees of freedom
## Residual deviance: 20831 on 828 degrees of freedom
## AIC: 26205
##
## Number of Fisher Scoring iterations: 4options(na.action = "na.fail") # Required for dredge to run
# Run all possible model permutations and tank according to AIC values
model_dredge_d <- dredge(m_saturated_d, beta = F, evaluate = T, rank = AICc)## Fixed term is "(Intercept)"options(na.action = "na.omit") # set back to default
head(model_dredge_d)## Global model call: glm(formula = dELA ~ debris_cov + morph_type + orientation +
## length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min +
## Elev_max + Elev_mean, family = poisson, data = glacier_scaled)
## ---
## Model selection table
## (Int) dbr_cov Elv_max Elv_men Elv_min lng mrp_typ orn P_not_mns
## 16384 4.606 -0.2407 0.1673 -0.1754 0.2705 0.05331 -0.02529 + -0.05807
## 8192 4.607 -0.2391 0.1667 -0.1750 0.2699 0.05355 -0.02607 + -0.05847
## 14336 4.607 -0.2391 0.1667 -0.1750 0.2699 0.05355 -0.02607 + -0.05847
## 12288 4.607 -0.2391 0.1667 -0.1750 0.2699 0.05355 -0.02607 + -0.05847
## 16128 4.615 -0.2421 0.1661 -0.1765 0.2732 0.05669 -0.02816 + -0.01951
## 7936 4.616 -0.2405 0.1656 -0.1761 0.2727 0.05693 -0.02892 + -0.02006
## P_yer Slp_men Slp_min T_men_mea.yr T_men_mns T_men_not_mns df
## 16384 0.03890 0.1237 -0.03609 70640.0000 -2.667e+04 -4.419e+04 21
## 8192 0.03875 0.1236 -0.03582 0.6414 -4.445e-01 20
## 14336 0.03875 0.1236 -0.03582 -2.023e-01 4.012e-01 20
## 12288 0.03875 0.1236 -0.03582 -0.5360 7.365e-01 20
## 16128 0.1254 -0.03725 70240.0000 -2.652e+04 -4.394e+04 20
## 7936 0.1253 -0.03698 0.6898 -4.971e-01 19
## logLik AICc delta weight
## 16384 -13081.53 26206.2 0.00 0.646
## 8192 -13084.36 26209.7 3.56 0.109
## 14336 -13084.36 26209.7 3.56 0.109
## 12288 -13084.36 26209.7 3.56 0.109
## 16128 -13085.96 26212.9 6.75 0.022
## 7936 -13088.75 26216.4 10.24 0.004
## Models ranked by AICc(x)nrow(model_dredge_d) # 16384 models in total## [1] 16384top_model_d <- get.models(model_dredge_d, subset = 1)[[1]]
top_model_d##
## Call: glm(formula = dELA ~ debris_cov + Elev_max + Elev_mean + Elev_min +
## length + morph_type + orientation + P_not_monsoon + P_year +
## Slope_mean + Slope_min + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + 1, family = poisson, data = glacier_scaled)
##
## Coefficients:
## (Intercept) debris_cov Elev_max Elev_mean
## 4.606e+00 -2.407e-01 1.673e-01 -1.754e-01
## Elev_min length morph_type orientationN
## 2.705e-01 5.331e-02 -2.529e-02 -3.527e-02
## orientationNE orientationNW orientationS orientationSE
## -1.955e-02 -1.892e-02 2.311e-01 5.327e-02
## orientationSW orientationW P_not_monsoon P_year
## 2.150e-01 1.781e-01 -5.807e-02 3.890e-02
## Slope_mean Slope_min T_mean_mea.yr T_mean_monsoon
## 1.237e-01 -3.609e-02 7.064e+04 -2.667e+04
## T_mean_not_monsoon
## -4.419e+04
##
## Degrees of Freedom: 848 Total (i.e. Null); 828 Residual
## Null Deviance: 26060
## Residual Deviance: 20830 AIC: 26210# Summarize top model
summary(top_model_d)##
## Call:
## glm(formula = dELA ~ debris_cov + Elev_max + Elev_mean + Elev_min +
## length + morph_type + orientation + P_not_monsoon + P_year +
## Slope_mean + Slope_min + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + 1, family = poisson, data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -13.0002 -3.8463 -0.6707 2.7638 23.1736
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.606e+00 1.829e-02 251.793 < 2e-16 ***
## debris_cov -2.407e-01 1.596e-02 -15.084 < 2e-16 ***
## Elev_max 1.673e-01 9.120e-03 18.340 < 2e-16 ***
## Elev_mean -1.754e-01 1.372e-02 -12.788 < 2e-16 ***
## Elev_min 2.705e-01 1.196e-02 22.623 < 2e-16 ***
## length 5.331e-02 6.913e-03 7.712 1.24e-14 ***
## morph_type -2.529e-02 5.659e-03 -4.469 7.84e-06 ***
## orientationN -3.527e-02 1.248e-02 -2.825 0.00473 **
## orientationNE -1.955e-02 1.404e-02 -1.392 0.16378
## orientationNW -1.892e-02 1.560e-02 -1.213 0.22519
## orientationS 2.311e-01 1.363e-02 16.960 < 2e-16 ***
## orientationSE 5.327e-02 1.687e-02 3.158 0.00159 **
## orientationSW 2.150e-01 1.482e-02 14.504 < 2e-16 ***
## orientationW 1.781e-01 1.367e-02 13.029 < 2e-16 ***
## P_not_monsoon -5.807e-02 1.363e-02 -4.259 2.05e-05 ***
## P_year 3.890e-02 1.308e-02 2.975 0.00293 **
## Slope_mean 1.237e-01 5.366e-03 23.052 < 2e-16 ***
## Slope_min -3.609e-02 4.563e-03 -7.909 2.59e-15 ***
## T_mean_mea.yr 7.064e+04 2.970e+04 2.379 0.01737 *
## T_mean_monsoon -2.667e+04 1.121e+04 -2.379 0.01737 *
## T_mean_not_monsoon -4.419e+04 1.858e+04 -2.379 0.01737 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 26056 on 848 degrees of freedom
## Residual deviance: 20831 on 828 degrees of freedom
## AIC: 26205
##
## Number of Fisher Scoring iterations: 4The most important fixed effects where mean annual temperature and mean temperature during monsoon phases. A more detailed description and visualization can be found in the dot and whisker plot at the end of this section.
# psudo R^2
r.squaredGLMM(top_model_d) # 83.01%## Warning: The null model is correct only if all variables used by the original
## model remain unchanged.## R2m R2c
## delta 0.8482790 0.8482790
## lognormal 0.8489160 0.8489160
## trigamma 0.8476367 0.8476367plot(top_model_d)
The residual vs fitted plot shows a mostly flat and even distribution of points, with greater deviances seen at the lower spectrum of points, likely due to the negative skew described above
When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Following an ad hoc transformation of the dependent variable ELA, this observed trend only showed a very marginal improvement, so I decided to keep ELA in its un-transformed state.
# Variable effect size visualization
ggcoefstats(x = top_model_d, conf.int = TRUE,
conf.level = 0.95, conf.method = "HPDinterval", exclude.intercept = T, stats.labels =F)
In the dot-whisker visual representation of the effect sizes and standard errors of our the variables we can see that mean elevation and mean annual temperature are the two most influential variables at predicting dELA (although the are also accompanied by large standard errors). These are then followed by precipitation the various available elevation variables, as well as the debris cover, which was already found to be relevant in predicting dELA in the hierarchical GLM and baysian models.
Compairisons and conclusions
(BIC_ELA <- BIC(m_lme4, m_bays, m_pca, top_model) %>%
arrange(BIC))## df BIC
## top_model 18 9119.323
## m_lme4 15 9657.006
## m_bays 15 9723.412
## m_pca 7 9960.545(BIC_dELA <- BIC(m_lme4_d, m_bays_d, m_pca_d, top_model_d) %>%
arrange(BIC))## df BIC
## m_bays_d 15 6308.751
## top_model_d 21 26304.696
## m_lme4_d 14 26733.776
## m_pca_d 6 28504.261# initial AIC ranking that I abandoned in favour of using BIC
#AICc_df <- AICc(m_lme4, m_lme4_d, m_bays, m_bays_d, m_pca, m_pca_d, top_model, top_model_d) %>%
# arrange(AICc)
#AICc_dfFor the between model comparison, I decided to use the Bayesian Information Criterion (BIC), as opposed to the Akaike`s information criterion (c). Using the BIC comes with a number of advantages such as assessing parsimony of different model types and model structures more consistently, and being more tolerant of models with slight under-fitting (likely to be the case in some of my models, as residual deviation was sometimes large at low and high end of the distribution). Furthermore BIC is more selective for the total number of variables included in the model and has a stronger negative weight associated to number of parameters. Applying greater penalization to models with many variables seems like a reasonable approach for this analysis, as all models had large sets of included variables, increasing the risk of over-fitting. A final argument for using BIC is that this has the potential to reduce the arbitrary methodological advantage of the automated variable selection methods, As the automated band selection methods optimize for AIC. Using BIC as the final assessment criterion allows for a slightly fairer comparison between the models. As there is a slight differentiation between AIC and BIC, this means that the automated selection models are not fully optimized for the assessment criterion.
Based on a compared BIC ranking between the models constructed under my four modeling approaches, it is seen that automated variable selection performed best for predicting both the current ELA. While I choice BIC as the final assessment criterion, it still unsurprising that the automated band selection methods performed the best, as they where optimized to have low parsimony. Despite the automated variable selection method practically being the “best” model, in terms of my personal confidence in the found results and overall reproduceability, I would favor the Bayesian model. The Bayesian models yielded an acceptably low BIC values for both ELA, and comes with the conceptual advantage of having its variables selected based on a priori knowledge/guesses of what makes mechanistic sense to include. The reason that the Bayesian model for dELA is likely to have a vastly lower BIC is that it may handle being defined as a Poisson distribution model differently than the frequentest approaches. It is possible that my initial choice of using a Poisson as opposed to a inverse Gaussian was sub-optimal, therefor resulting in the inflated BIC values. While unlikely based on my understanding of how BIC is calculated, an alternative explanation could be that when using a Poisson distribution the underlying math for calculating BIC, and that the inflated BIC values are a calculation artifact and not a result of using the wrong distribution for the dELA models.
Regardless of what modeling approach is preferred, synthesizing the results of all my models helps provide more robust answers to the main research questions.
1. Which are the most important topographical and climatological drivers of glacier equilibrium line altitude (ELA)?
Synthesizing the results from all my modeling approaches, (while favoring my a priori defined hierarchical models), I have identified mean annual temperature and mean elevation and as the two most important variables for predicting ELA. Despite the automated selection method choosing temperature in non monsoon phases over mean annual temperature, I believe that is likely an artifact of high degree of co-linearity between the variables and the automated selection method being highly sensitive to marginal deviation in effect size. Overall though, both Philosophical modeling approaches converge on the fact that temperature values averaged across longer timescales are critical to predicting ELA. Mean precipitation in non monsoon periods and temperature variability are generally the next two most important variables and where consistently important predictors across all modeling approaches. Overall, when considering the results of my models I conclude that mean annual temperature and mean elevation and as the two most important variables for predicting ELA, while mean precipitation in non monsoon periods and temperature variability being the two strongest key secondary variables.
2. Can we explain ELA changes since the Little Ice Age with topographical and climatological variables?
Based on the synthesized results of all my models I believe that ELA changes since the Little Ice Age can be reasonably predicted using climatological and topographical variables. As when attempting to predict ELA, all models converged on mean annual temperature being a critical variable when predicting dELA. Furthermore both the hierarchical and automated section models identified debris cover as a variable with large influence on dELA. Additional variables that both modeling approaches identified as being particularly useful for predicting dELA where precipitation in non monsoon periods, and the various available elevation metrics. Overall, when considering the results of my models I conclude that mean annual temperature and debris cover and as the two most important variables for predicting dELA, while mean precipitation in non monsoon periods and catchment elevation being the two strongest key secondary variables.